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## 2012.72: Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number

2012.72: Awad H. Al-Mohy, Nicholas J. Higham and Samuel D. Relton (2012) Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number.

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## Abstract

The most popular method for computing the matrix logarithm is the inverse scaling and squaring method, which is the basis of the recent algorithm of [A. H. Al-Mohy and N. J. Higham, \emph{Improved inverse scaling and squaring algorithms for the matrix logarithm}, SIAM J. Sci.\ Comput., 34 (2012), pp.~C152--C169]. We show that by differentiating the latter algorithm a backward stable algorithm for computing the Fr\'echet derivative of the matrix logarithm is obtained. This algorithm requires complex arithmetic, but we also develop a version that uses only real arithmetic when $A$ is real; as a special case we obtain a new algorithm for computing the logarithm of a real matrix in real arithmetic. We show experimentally that our two algorithms are more accurate and efficient than existing algorithms for computing the Fr\'echet derivative. We also show how the algorithms can be used to produce reliable estimates of the condition number of the matrix logarithm.

Item Type: MIMS Preprint matrix logarithm, principal logarithm, inverse scaling and squaring method, Fr\'{e}chet derivative, condition number, Pad\'{e} approximation, backward error analysis, matrix exponential, matrix square root, MATLAB, logm. MSC 2000 > 15 Linear and multilinear algebra; matrix theoryMSC 2000 > 65 Numerical analysis 2012.72 Nick Higham 25 July 2012